Takiff algebras with polynomial rings of symmetric invariants

TL;DR

This paper proves that Takiff algebras of certain Lie algebras retain polynomial rings of symmetric invariants, extending previous results and under mild restrictions on the original algebra.

Contribution

It establishes that the polynomial ring property of symmetric invariants is preserved in Takiff algebras under mild conditions, extending prior work by Rais-Tauvel, Macedo-Savage, and Arakawa-Premet.

Findings

Takiff algebras of Lie algebras with polynomial symmetric invariants also have polynomial symmetric invariants.

The result generalizes previous theorems to broader classes of Lie algebras.

Under mild restrictions, the polynomial nature of symmetric invariants is preserved in Takiff constructions.

Abstract

Extending results of Rais-Tauvel, Macedo-Savage, and Arakawa-Premet, we prove that under mild restrictions on the Lie algebra $\mathfrak q$ having the polynomial ring of symmetric invariants, the m-th Takiff algebra of $\mathfrak q$, $\mathfrak q\langle m\rangle$, also has a polynomial ring of symmetric invariants.